Poincar\'e formula PoincareFormula 2003-06-09 04:43:01 2007-09-07 18:30:02 Theorem % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Let $K$ be finite oriented simplicial complex of dimension $n$. Then $$\chi(K)= \sum_{p=0}^n (-1)^p R_p(K),$$ where $\chi(K)$ is the Euler characteristic of $K$, and $R_{p}(K)$ is the $p$-th Betti number of $K$. This formula also works when $K$ is any finite CW complex. The Poincar\'e formula is also known as the Euler-Poincar\'e formula, for it is a generalization of the Euler formula for polyhedra. If $K$ is a compact connected orientable surface with no boundary and with genus h, then $\chi(K)=2-2h$. If $K$ is non-orientable instead, then $\chi(K)=2-h$.